You can suck most of the ink out of the cartridge with your syringe and blunt tip needle, then rinse with water in the same manner. Inject Distilled water with your syringe, shake it around and suck it out, repeat until clean. To clean out the exit valve you will have to depress the valve itself, you could use the end of the syringe (without a needle attached), with the air vent hole OPEN, and suck out the distilled water until it’s clean.
The other option would be to use the 2 sets of cartridges, so when one runs out you have a full cart on hand without having to stop and refill. This is what I do to keep things going with less down time, yes I have to fill an entire set of 2 carts at once, but that is much less time consuming then 1 set at a time.